Why is that the theory of automorphic forms concentrates on the case of half-integral weight? I read in Borel's book "Automorphic forms on $SL_2$" (Section 18.5) that by considering the finite or universal coverings of $SL_2$, one can define modular forms with rational or even real weight. Borel mentions in passing that the case of half-integral weight is particularly important. Why?!

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Since the other MO answer linked-to in Alison Miller's comment did not mention this (although Alison M. did make a comment in this direction there): indeed, the Segal-Shale-Weil/oscillator repn produces representations and automorphic forms for two-fold covers of $Sp_{2n}$, but not for more general covers. This gives the classical theta correspondences, which accounted for nearly all the "early" examples of Langlands functoriality. The technical point seems to be that the two-fold cover of $Sp_{2n}$ admits a very small "minimal" repn, which then usefully decomposes (see also "Howe correspondence") over mutually commuting subgroups ("dual reductive pairs").

I may be mistaken, but it is my impression that other covers do not have such small "minimal" repns, so that the analogues of theta correspondences are either not present at all, or don't have the multiplicity-one properties.

Another key-phrase is "first occurrence" in work of Kudla-Rallis and collaborators, examining in precise detail the "theta correspondences"' repn-theoretic properties. I am not aware of any comparable successes for other covers. Nevertheless, there is work of Brubaker-Bump-Friedberg on "multiple Dirichlet series" that looks for applications of more general covers.

I think that modular forms (for $SL_2$) of integer and half-integer weights are most important for arithmetic, while modular forms of other (real or complex) weights are primarily objects of analytic interest. I admit, modular forms of other weights might have connections to arithmetic -- I don't know anything about harmonic Maass forms, VOA theory, etc., so I can't rule out a connection. But the most important connection to arithmetic -- the Euler product -- seems absent outside of half-integer and integer weights.

The reason for this is the "metaplectic kernel" and a good reference is the paper "Computation of the Metaplectic Kernel" by Gopal Prasad and Andrei Rapinchuk, in Inst. Hautes Études Sci. Publ. Math. No. 84 (1996), 91–187 (1997).

To have a good theory of Euler products, the modular forms should correspond to automorphic representations of some metaplectic group. So, for $SL_2$, you need a central extension of locally compact groups:
$$1 \rightarrow U(1) \rightarrow \tilde G_{\mathbb A} \rightarrow G_{\mathbb A} \rightarrow 1,$$
where $U(1)$ is the circle, and you also need a splitting of the extension over $G_{\mathbb Q}$.

When $G = SL_2$, the only nontrivial such extensions with splittings come from two-fold extensions of $SL_2({\mathbb A})$ -- the traditional metaplectic group. So you only see modular forms of half-integer weight if you want to work adelically, decompose automorphic representations into local pieces, get an Euler product, etc..

For other groups over other fields, you can get other sorts of central extensions. This is the subject of Prasad-Rapinchuk's paper cited above.

Personally, I prefer an even more restrictive class of "metaplectic groups" which arise from the algebraic framework of Brylinski and Deligne (central extensions of reductive groups by $K_2$).